As far as I know the most common name is "hink pink", if anyone wants to look this up (or sometimes "hinky pinky"). Here's a 1981 book, https://archive.org/details/hinkpinkbookorwh00burn/ and here's a short description from the 50s, https://archive.org/details/realbookofgames0000unse/page/134... Searching further, apparently Hink Pink was the name of an 18th century pirate ship; I'm not sure if there's a relation to the game.
According to this 1941 Life Magazine issue, teenage girls in Atlanta were making up rhyming pairs like this at the time under the name "stinky pinky". https://archive.org/details/Life-1941-01-27-Vol-10-No-4/mode... Webster's Dictionary from the 60s has the game listed under that name, https://archive.org/details/webstersthirdnew0000phil_l0b1/mo... and that name also seems to continue to today, e.g. by the radio show Loveline.